Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA, and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.
J Chem Phys. 2017 Aug 14;147(6):064112. doi: 10.1063/1.4995301.
The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics-as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model-can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation-because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation-it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton's equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrödinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton's equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton's equations (and presumably also in the Schrödinger equation) can cause very significant errors.
迈尔-米勒（MM）经典振子（电子+核）哈密顿量用于非绝热动力学，例如，最近开发的对称准经典（SQC）窗口模型，可以用电子自由度的绝热或非绝热表示来表示，两者是彼此的正则变换，从而给出相同的动力学。尽管最近对这种 SQC/MM 方法的应用都是在绝热表示中进行的，因为大多数具有可比较的精确量子结果的基准模型问题通常都是在绝热表示中定义的，但在绝热表示中工作通常会更加方便，例如，当使用 Born-Oppenheimer 势能面（PES）和衍生耦合时，这些耦合来自电子结构计算。然而，从绝热 MM 哈密顿量得出的运动方程（EOM）（即哈密顿方程）除了常见的一阶导数耦合外，还涉及二阶非绝热耦合项（就像量子薛定谔方程一样），而后者计算起来要困难得多。因此，本文重新研究了 MM 哈密顿量的绝热版本，并描述了对经典绝热 EOM 的修改，这些修改完全等效于哈密顿方程，但不涉及二阶导数耦合。二阶导数耦合项并没有被忽略；它们只是不出现在这些修正的绝热 EOM 中。这意味着可以在不进行近似的情况下，在绝热表示中进行 SQC/MM 计算，只需要 PES 和一阶导数耦合元素。本文还给出了 SQC/MM 计算的示例结果，说明了这一点，以及忽略哈密顿方程（大概还有薛定谔方程）中的二阶导数耦合会导致非常大的误差的事实。
